Analysis of Contact Stresses and Rolling Resistance of Truck-Bus Tyres Under Diﬀerent Working Conditions

sustainability

Article Analysis of Contact Stresses and Rolling Resistance of Truck-Bus Tyres under Diﬀerent Working Conditions

Minrui Guo 1,2,*, Xiangwen Li 1, Maoping Ran 1, Xinglin Zhou 1 and Yuan Yan 1

1 College of Automotive and Transportation Engineering, Wuhan University of Science and Technology, Wuhan 430065, China; [email protected] (X.L.); [email protected] (M.R.); [email protected] (X.Z.); [email protected] (Y.Y.) 2 College of Mechanical and Energy Engineering, Huanghuai University, Zhumadian 463000, China * Correspondence: [email protected]

Received: 13 October 2020; Accepted: 15 December 2020; Published: 18 December 2020

Abstract: In this work, to analyse the changing characteristics of contact stresses in the tyre–pavement interface and the functional relationship between rolling resistance and the working conditions of truck-bus tyres, a three-dimensional tyre–pavement model is established and used to predict the distribution of contact stresses and rolling resistance under diﬀerent working conditions of the tyre, comprising various tyre loads, inﬂation pressures, and velocities. Results show that the magnitude relationship between transverse and longitudinal contact stresses is related to rolling conditions, and overload and low tyre pressure are important contributors to the wear of the tyre shoulder. In addition, the proposed exponential equation presents a method that can be used to forecast rolling resistance related to the working conditions of the truck-bus tyre, and a similar method can be used to predict the rolling resistances of other types of tyres.

Keywords: contact stresses; rolling resistance; braking; free rolling; load; inﬂation pressure; speed

1. Introduction Different simulation studies [1,2] and experimental results [3,4] have shown that contact stresses between tyre and pavement not only have a great influence on vehicle handling and stability but also significantly affect the prediction of pavement responses and performance [5–9]. Regardless of the type of tyre or pavement, the contact stresses or forces in the tyre–pavement interface can be resolved into three orthogonal directions, namely, longitudinal (X), lateral (Y), and vertical (Z) contact stress or force distributions [10], as shown in Figure1. The directions of the coordinate systems adopted by different countries and professions may be different. In this paper, the orientation of these contact stresses is defined according to the Society of Automotive Engineers’ (SAE) coordinate system [11], or in accordance with the right-hand rule. In addition, the tyre rolling resistance (RR) is responsible for 20% of the fuel consumption of truck tyres [12], and RR predictions generate better estimations of fuel consumption and greenhouse gas emissions [13]. Previous studies have demonstrated that the vertical contact stress of the tyre–road interface has a non-uniform distribution, and horizontal contact stress are developed [14]. Some studies have regarded vertical contact stress as uniformly distributed [15–17], some have only considered non-uniformly distributed vertical stress and ignored tangential stress [18–20], and some have obtained the distribution of the three orthogonal contact stresses under both static and vehicle manoeuvring conditions [21–23]. However, the change trends of contact stresses during rolling conditions have not been studied and depend on many factors, including the tyre structure (geometry, tread pattern, rubber constitutive model and reinforcement), pavement surface conditions, tyre working conditions (tyre load, inflation pressure, and speed) and rolling state (accelerating, free rolling, braking). Cho [24]

Sustainability 2020, 12, 10603; doi:10.3390/su122410603 www.mdpi.com/journal/sustainability Sustainability 2020, 12, x FOR PEER REVIEW 2 of 17 Sustainability 2020, 12, 10603 2 of 16 conditions, tyre working conditions (tyre load, inflation pressure, and speed) and rolling state (accelerating, free rolling, braking). Cho [24] predicted the RR of a patterned tyre (205/60R15) from predicted the RR of a patterned tyre (205/60R15) from a passenger car by utilising static tyre contact a passenger car by utilising static tyre contact analysis. Aldhufairi [25] investigated the RR of a analysis. Aldhufairi [25] investigated the RR of a radial passenger-car tyre (225/55R17) based on material radial passenger-car tyre (225/55R17) based on material viscoelasticity. Rafei [26] implemented a viscoelasticity. Rafei [26] implemented a simulation of the RR of a passenger-car tyre (185/65R14) using simulation of the RR of a passenger-car tyre (185/65R14) using the finite element method. Ghost [12] the ﬁnite element method. Ghost [12] analysed the RRs of truck-bus tyres with a carbon black tread as analysed the RRs of truck-bus tyres with a carbon black tread as well as a silica tread under rated well as a silica tread under rated conditions. However, variation of the RR of truck-bus tyres under conditions. However, variation of the RR of truck-bus tyres under different working conditions and diﬀerent working conditions and the functional relationship between RR and tyre working conditions the functional relationship between RR and tyre working conditions have not been studied. have not been studied.

Figure 1. ThreeThree orthogonal directions of contact stresses.

Some scholarsscholars have have used used experimental experimental equipment equipm toent measure to measure contact contact stresses stresses and RR. and For example,RR. For Anghelacheexample, Anghelache et al. [27] used et al. 30 [27] strain used gauged 30 strain sensing gauged elements sensing to measure elements the tri-axial to measure stress the distribution tri-axial stressof a 205 distribution/55R16 tyre of ofa 205/55R16 a passenger tyre car of witha passeng a 240er kPa car with inflation a 240 pressure kPa inflation at a speedpressure of 0.54at a speed km/h. Anghelacheof 0.54 km/h. et al.Anghelache [28] used theet sameal. [28] measuring used the device same to measuremeasuring the device stress distributionto measure of the a 11R22.5 stress distributiontruck tyre with of a a 11R22.5 780 kPa truck inflation tyre pressurewith a 780 at kPa a speed inflation of 3.2 pressure km/h. De at Beera speed and of Fisher 3.2 km/h. [29] De applied Beer and63 strain Fisher measurement [29] applied channels 63 strain to measurement capture the tri-axial channe tyre-roadls to capture interaction, the tri-axial and the tyre-road working interaction, conditions andof the the 12R22.5 working truck conditions tyre were of the 20 and12R22.5 25 kN truck vertical tyre were loads 20 with and a 25 520 kN kPa vertical inflation loads pressure with a at520 a kPafree rollinginflation speed pressure of 1.08 at kma free/h. Chenrolling [30 speed] applied of 1.08 a pressure-sensitive km/h. Chen [30] filmappl toied determine a pressure-sensitive the contact filmregion to anddetermine stress distributionthe contact region between and a 215stress/75R15 distribution tyre of a between light vehicle a 215/75R15 and asphalt tyre pavementof a light vehicleunder static and conditions.asphalt pavement Ghost [ 12under] investigated static conditions. the RR values Ghost of [12] three investigated tyres in drum-type the RR RRvalues testing of threeequipment, tyres in and drum-type a good correlation RR testing betweenequipment, the and simulated a good and correlation measured between RR values the simulated was observed. and measuredHowever, theRR costvalues of experimental was observed. equipment However, is high, the thecost installation of experimental is complex equipment and time-consuming, is high, the installationand substantial is complex amounts and of manpower time-consuming, and material and substantial resources are amounts required. of Inmanpower addition, and many material factors resourcesaffect the accuracyare required. and reliability In addition, of the experimentalmany factors test affect results, the including accuracy sensor and aspects reliability (the number,of the experimentaldirection, and arrangement),test results, including tyre aspects sensor (the aspect type, structure,s (the number, materials, direction, working and conditions, arrangement), and rolling tyre aspectsstate), and (the the type, road structure, aspect. During materials, the testingworking process, conditions, due to and the dirollingfferent state), combinations and the ofroad measured aspect. variablesDuring the and testing because process, the actual due drivingto the different conditions combinations of a tyre are diofffi measuredcult to achieve, variables it is veryand because challenging the actualto take driving into account conditions all the of working a tyre are conditions difficult ofto theachieve, tyre and it is achievevery challenging different driving to take conditions.into account all theBased working on preceding conditions analysis, of the tyre this and study achieve primarily different develops driving a conditions. theoretical methodology for the predictionBased ofon thepreceding contact analysis, stresses andthis RRstudy of primarily the truck-bus develops radial a tyretheoretical (275/70R22.5) methodology under for rolling the predictionconditions of and the explores contact the stresses change and rule RR of of tri-axial the truck-bus contact stressesradial tyre and (275/70R22.5) the functional under relationship rolling conditionsbetween RR and and explores tyre working the change conditions. rule of tri-axial The findings contact provide stresses valuable and the functional information relationship about the betweendistribution RR ofand contact tyre working stresses, theconditions. location The of tyre findings wear, andprovide the functionalvaluable information expression ofabout tyre RR,the distributionespecially in of terms contact of tyre stresses, and pavement the location contact of ty mechanicsre wear, and and the fuel-economic functional tyreexpression production of tyre/design. RR, especially in terms of tyre and pavement contact mechanics and fuel-economic tyre 2. Establishment and Verification of Tyre–Pavement Finite Element Model production/design. Two-dimensional (2D) and three-dimensional (3D) tyre models can be used to solve tyre dynamics issues. Simplified 2D models have been used in tyre dynamics analysis to forecast the characteristics and handling stability of a tyre [31]. For example, the classical point contact model can simplify the tyre into a spring-damping system to analyse the vertical forces acting on the tyre [32]. Although 2D tyre models can solve many problems in tyre mechanics and have some successful applications, Sustainability 2020, 12, 10603 3 of 16 these simplified models are not suitable for the prediction and calculation of the contact stresses between the tyre and pavement. They also cannot be used to analyse the complex composition of each part of the tyre in detail or the nonlinear characteristics of tyre materials [22]. In this study, a complete 3D tyre model that considers the actual dimensions and nonlinear characteristics of the tyre is established with the assistance of ABAQUS software.

2.1. The Constitutive Model of Rubber Material Rubber material has a hyperelastic property that exhibits an obvious nonlinear response, and the rubber constitutive model is expressed in terms of the strain potential energy. The general expression of this model is as follows:

N N X i j X 1 2i U = Cij(I1 3) (I2 3) + (J 1) (1) − − Di − i+j=1 i=1 where U is strain energy, Cij is the constant of material properties, N is the polynomial order, I1 and I2 are the distortion measurement of materials, Di is the material parameter that introduces the compression feature, and J is the bulk modulus. When N = 1, Equation (1) is rewritten as:

1 2 U = C10(I1 3) + C01(I2 3) + (J 1) (2) − − D1 −

Equation (2) is the Mooney–Rivlin material model. When C01 = 0, the Mooney–Rivlin material model is transformed into the Neo-Hookean model, namely:

1 2 U = C10(I1 3) + (J 1) (3) − D1 −

The Neo-Hookean material model can predict both the moderate and small strain of rubber material and is characterised by good reliability and stability. Therefore, Equation (3) is applied to analyse the hyperelasticity characteristics of rubber material. In addition, rubber material also has viscoelastic properties, which are usually expressed by Prony series parameters.

2.2. Rolling Resistance of the Tyre The RR is equal to the energy loss per revolution of rolling divided by the distance of each revolution [33]. The energy loss of each element volume of tyre material in each cycle is described by Equation (4). 2 ∆W = π E0 ε tan δ (4) · · 0· E00 tan δ = (5) E0 The rolling energy loss of the tyre in each cycle is then as follows: X ELoss = ∆W v (6) i· i i

Finally, the calculation expression of the RR is as follows: P ∆Wi vi E i · RR = Loss = (7) 2πr 2πr Sustainability 2020, 12, 10603 4 of 16

Sustainability 2020, 12, x FOR PEER REVIEW 4 of 17 where ∆W represents the energy loss of each element, E00 and E0 represent storage and loss moluli, ε0 represents strain amplitude, tan δ represents the loss factor,'' vi represents' the volume of element i of where ΔW represents the energy loss of each element, E and E represent storage and loss tyre material, ELossε represents total energy loss, andδ r represents the tyre rolling radius. moluli, 0 represents strain amplitude, tan represents the loss factor, vi represents the E 2.3. Developmentvolume of element a Complete i of tyre 3D Tyrematerial, Model Loss represents total energy loss, and r represents the tyre rolling radius. Figure2 presents the tyre–pavement contact model and the components of a complex radial tyre (275/70R22.5),2.3. Development namely of a Complete a truck-bus 3D Tyre Model radial tyre containing four longitudinal grooves, and each tyre componentFigure is distinguished2 presents the tyre–pavement by a diﬀerent contact colour. model and The the outer components diameter of a complex is 958 mm,radial the cross- tyre (275/70R22.5), namely a truck-bus radial tyre containing four longitudinal grooves, and each sectional width is 276 mm, and steel-belt-1 and steel-belt-2 have orientation angles of 20◦and 20◦, tyre component is distinguished by a different colour. The outer diameter is 958 mm, the cross- − respectively.sectional The lengthwidth is of276 the mm, mesh and steel-belt-1 is 16 mm, and and steel-belt-2 the width have is orientation 9 mm in theangles contact of 20°and area. −20°, Steel- belt-1 and steel-belt-2respectively. are locatedThe length approximately of the mesh is 16 15 mm, and and 17 the mm width away is 9 frommm in the the outercontactsurface area. Steel- of the tread, respectively.belt-1 To and optimise steel-belt-2 computational are located approximately eﬃciency 15 and and ensure 17 mm betteraway from convergence, the outer surface the tread of the of the tyre in contacttread, with respectively. the road surface To optimise is selected computational as a dense efficiency mesh, and and ensure the better other convergence, non-contact the locationstread of the of the tyre in contact with the road surface is selected as a dense mesh, and the other non-contact tyre are selectedlocations as of athe sparse tyre are mesh. selected The as a material sparse mesh parameters. The material of parameters the tyre compositionof the tyre composition are based on the authors’ previousare based on study the authors’ [34]. previous study [34].

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 2. FigureMeshes 2. Meshes of tyre of tyre components. components. (a ()a Tread) Tread with with four longitudinal four longitudinal grooves; (b grooves;) sidewall; ((bc)) sidewall; steel-belt-1; (d) steel-belt-2; (e) bead; (f) carcass; (g) full tyre; (h) the contact model of (c) steel-belt-1; (d) steel-belt-2; (e) bead; (f) carcass; (g) full tyre; (h) the contact model of tyre–pavement. tyre–pavement. 2.4. Modelling Tyre-Pavement Contact 2.4. Modelling Tyre-Pavement Contact Complex,Complex, non-linear non-linear contact contact problems problems must must be consideredbe considered in in the the studystudy ofof tyre–pavementtyre–pavement contact. There arecontact. many There factors arethat many prevent factors that successful prevent su modellingccessful modelling of tyre–pavement of tyre–pavement contact; contact; for example, some challengesexample, thatsome arisechallenges when that modelling arise when modell withing atwo-solid with a two-solid mode mode include include the the non-linear contact contact characteristics of the tyre–pavement, material characteristics of the tyre, fast moving characteristics of the tyre–pavement, material characteristics of the tyre, fast moving conditions, great deformation of the tyre, and the nonlinear frictional relationship between the friction coefficient and velocity [35]. It is clear that dealing with the tyre–pavement problem with a two-solid mode is not sufficient. In this study, when the tyre was in contact with the road surface, the degree of the deflection of the tyre was much larger than that of the pavement. It can be presumed that the pavement is a non-deformable rigid surface to achieve better astringency and stability, and the accuracy and reliability of contact stresses analysis can still be guaranteed. In previous research, the pavement has Sustainability 2020, 12, x FOR PEER REVIEW 5 of 17 Sustainability 2020, 12, x FOR PEER REVIEW 5 of 17 Sustainability 2020, 12, 10603 5 of 16 conditions, great deformation of the tyre, and the nonlinear frictional relationship between the frictionconditions, coefficient great anddeformation velocity [35].of the It istyre, clear and that the dealing nonlinear with thefrictional tyre–pavement relationship problem between with the a two-solidfriction coefficient mode is not and sufficient. velocity [35].In this It isstudy, clear whthaten dealing the tyre with was the in tyre–pavementcontact with the problem road surface, with a been consideredthetwo-solid degree as amodeof rigid the is deflection not surface sufficient. thatof the In can thistyre successfully study,was much when larger the capture tyre than was that contactin contactof the stresses withpavement. the [road22 It, 36 cansurface,–38 be]. The 3D model of tyre–pavementpresumedthe degree that of thethe contact pavementdeflection is shown isof a thenon-deformable tyre in Figure was much2 h.rigid larger surface than to thatachieve of the better pavement. astringency It can and be stability,presumed and that the the accuracy pavement and is reliability a non-deformable of contact rigid stresses surface analysis to achieve can still better be astringencyguaranteed. andIn 2.5. Validationpreviousstability, of the research,and 3D Contactthe accuracythe pavement Model and reliability has been of consid contactered stresses as a rigidanalysis surface can stillthat becan guaranteed. successfully In captureprevious contact research, stresses the [22,36–38]. pavement The has 3D been model consid of tyre–pavementered as a rigid contact surface is shown that can in Figure successfully 2h. The tyre–pavementcapture contact interactionstresses [22,36–38]. model The can 3D bemodel validated of tyre–pavement by the tyre contact load–displacement is shown in Figure 2h. relationship in a static condition.2.5. Validation Figure of the 3D3 Contactpresents Model the calculation comparisons of tyre deflections at a tyre load 2.5. Validation of the 3D Contact Model of 25 kN and diTheff erenttyre–pavement inflation interaction pressures. model When can thebe validated tyre load by wasthe constant,tyre load–displacement the lower the tyre The tyre–pavement interaction model can be validated by the tyre load–displacement inflation pressure,relationship the in greatera static condition. the displacement. Figure 3 presents The the tyre calculation loaded comparisons measurement of tyre systemdeflections is at presented arelationship tyre load of in 25 a kNstatic and condition. different Figure inflation 3 presents pressures. the Whencalculation the tyre comparisons load was constant,of tyre deflections the lower at in Figure4athea and tyre tyre wasload inflation usedof 25 pressure,kN to and measure different the greater the inflation deformation the displa pressucement.res. data When The of thetyre tyres tyre loaded load with measurement was di constant,fferent system loads.the lower is Figure 4b shows comparisonspresentedthe tyre inflation in between Figure pressure, 4a calculatedand was the used greater and to measurethe measured displa thecement. deformation tyre The displacements tyre data loaded of tyres measurement atwith di ffdifferenterent system tyreloads. loadsis and an inflationFigurepresented pressure 4b shows in of Figure 530comparisons 4a kPa. and Whenwas between used the to calculated inflationmeasure anthe pressured deformationmeasured was tyre data displacements constant, of tyres with the at different largerdifferent theloads. tyre tyre load, Figure 4b shows comparisons between calculated and measured tyre displacements at different tyre the larger theloads displacement. and an inflation pressure Figure 4ofc 530 shows kPa. When comparisons the inflation at pressure di fferent was constant, tyre loads the larger and anthe inflation tyreloads load, and the an largerinflation the pressure displacement. of 530 FigurekPa. When 4c shows the inflation comparisons pressure at differentwas constant, tyre loads the larger and anthe pressure ofinflation 730tyre kPa.load, pressure the Figure larger of4 730thereveals displacement.kPa. thatFigure good 4 Figurereveals consistency 4c that shows good comparisons wasconsistency achieved at was different betweenachieved tyre between theloads calculated and the an and tested displacementscalculatedinflation pressureand at tested various of displacements 730 tyre kPa. loadsFigure at various and4 reveals inflation tyre that loads good pressures. and consistency inflation pressures. was achieved between the calculated and tested displacements at various tyre loads and inflation pressures.

(a) (b) (a) (b) Figure 3. ComparisonsFigure 3. Comparisons of deﬂections of deflections at a tyreat a tyre load load of 25of 25 kN kN and and inﬂationinflation pressures pressures of ( ofa) 530 (a) 530kPa kPa and andFigure (b) 7303. Comparisons kPa. of deflections at a tyre load of 25 kN and inflation pressures of (a) 530 kPa (b) 730 kPa. and (b) 730 kPa.

(a) Sustainability 2020, 12, x FOR PEER REVIEW (a) 6 of 17

(b) (c)

Figure 4. (Figurea) Tyre 4. ( statica) Tyre loaded static loaded test. test. (b )(b Comparisons) Comparisons between between calculated calculated and measured and measured tyre tyre displacements at different tyre loads and inflation pressures of 530 kPa and (c) 730 kPa. displacements at diﬀerent tyre loads and inﬂation pressures of 530 kPa and (c) 730 kPa. 3. Effects of Tyre Working Conditions on Contact Stresses

3.1. Simulation Condition Design of Tyre–Pavement Contact For a two-wheel group, an axle load of 100 kN is the Chinese standard in the design of asphalt and town pavement [39]; thus, in the simulation, the vertical load (F) of each tyre was 25 kN in accordance with the standard axle load, and the vertical overloads of 30% and 60% were 32.5 and 40 kN, respectively. The rated inflation pressure (P) for a 275/70R22.5 tyre was 730 kPa [40], and the inflation pressures of 530 and 930 kPa respectively correspond to low and high inflation pressures. In addition, the friction coefficient between the tyre and road surface was 0.6 [34].

3.2. Realisation of the Steady Rolling State Analysis of contact stresses requires not only detailed information about tyre materials and components but also knowledge of non-linear characteristics and dynamic analysis. When the tyre is in the breaking state, the direction of braking torque (T) is opposite to the orientation of tyre rolling, as shown in Figure 5. When the tyre is in a free-rolling state, there is no braking torque. Figure 6 shows the longitudinal reaction force at different angular velocities for a linear velocity of V = 15 km/h. At the moment, the angular velocity of the free-rolling tyre corresponds to the angular velocity at zero torque. When the tyre is in the breaking state, the braking torque is transmitted to the tyre, and the angular velocity is less than that in the free-rolling condition, as shown in Figure 6. When the longitudinal reaction force has a positive value, it means that the tyre is in the braking state. When the longitudinal reaction force is basically constant, it means that the tyre is under full-braking conditions. Because the braking and acceleration forces are similar, this study focuses on comparing the change characteristics of the contact stresses and contact area with basic working conditions under free-rolling and full-braking conditions. The basic working condition was a tyre load of 25 kN, an inflation pressure of 730 kPa, and a speed of 15 km/h. Based on the described research method and results, the functional relationship between RR and tyre working conditions are explored. Sustainability 2020, 12, 10603 6 of 16

3. Eﬀects of Tyre Working Conditions on Contact Stresses

3.1. Simulation Condition Design of Tyre–Pavement Contact For a two-wheel group, an axle load of 100 kN is the Chinese standard in the design of asphalt and town pavement [39]; thus, in the simulation, the vertical load (F) of each tyre was 25 kN in accordance with the standard axle load, and the vertical overloads of 30% and 60% were 32.5 and 40 kN, respectively. The rated inflation pressure (P) for a 275/70R22.5 tyre was 730 kPa [40], and the inflation pressures of 530 and 930 kPa respectively correspond to low and high inflation pressures. In addition, the friction coefficient between the tyre and road surface was 0.6 [34].

3.2. Realisation of the Steady Rolling State Analysis of contact stresses requires not only detailed information about tyre materials and components but also knowledge of non-linear characteristics and dynamic analysis. When the tyre is in the breaking state, the direction of braking torque (T) is opposite to the orientation of tyre rolling, as shown in Figure5. When the tyre is in a free-rolling state, there is no braking torque. Figure6 shows the longitudinal reaction force at diﬀerent angular velocities for a linear velocity of V = 15 km/h. At the moment, the angular velocity of the free-rolling tyre corresponds to the angular velocity at zero torque. When the tyre is in the breaking state, the braking torque is transmitted to the tyre, and the angular velocity is less than that in the free-rolling condition, as shown in Figure6. When the longitudinal reaction force has a positive value, it means that the tyre is in the braking state. When the longitudinal reaction force is basically constant, it means that the tyre is under full-braking conditions. Because the braking and acceleration forces are similar, this study focuses on comparing the change characteristics of the contact stresses and contact area with basic working conditions under free-rolling and full-braking conditions. The basic working condition was a tyre load of 25 kN, an inﬂation pressure of 730 kPa, and a speed of 15 km/h. Based on the described research method and results, the functional relationshipSustainabilitySustainability between 20202020,, 1212 RR,, xx FORFOR and PEERPEER tyre REVIEWREVIEW working conditions are explored. 77 ofof 1717

FigureFigureFigure 5. 5.5.Tyre Tyre force force diagram during during braking. braking.

FigureFigureFigure 6. Longitudinal 6.6. Longitudinal force force at at di differentﬀerent angular angular velocities velocities for for a linear a linear velocity velocity of V == of 1515 Vkm/h.km/h.= 15 km/h.

3.3.3.3. EffectEffect ofof TyreTyre LoadLoad onon thethe ChanChangege CharacteristicsCharacteristics ofof ContactContact StressesStresses The inflation pressure of 730 kPa and a tyre speed of 15 km/h were considered to be invariant, while the tyre load was varied to examine the effect of tyre load on tyre–pavement contact stresses. Figure 7 shows the vertical contact stresses (CPRESS) underunder freefree rollingrolling andand fullfull brakingbraking withwith tyretyre loadsloads ofof 2525 andand 4040 kN.kN. WithWith thethe gradualgradual increaseincrease of the tyre load under the condition of steady rolling,rolling, thethe shapeshape ofof thethe contactcontact regionregion waswas founfound to transition from approximately circular to rectangular,rectangular, andand thethe contactcontact areaarea alsoalso graduallygradually inincreased.creased. RegardlessRegardless ofof whetherwhether thethe tyretyre waswas inin thethe free-rollingfree-rolling oror full-brakingfull-braking condition,condition, thethe verticverticalal stressstress atat thethe treadtread ofof thethe tyretyre waswas foundfound toto changechange onlyonly slightly.slightly. However,However, thethe verticalvertical stressstress was found to increase at the tyre shoulder. When thethe tyretyre loadload waswas small,small, thethe maximummaximum valuevalue ofof vertverticalical contactcontact stressstress waswas nearnear thethe centrecentre ofof thethe tread,tread, whereaswhereas whenwhen thethe loadload becamebecame large,large, thethe maximum value of vertical contactcontact stressstress movedmoved towardtoward thethe tyretyre shoulder.shoulder. UnderUnder differentdifferent loadload conditions,conditions, thethe verticalvertical stressstress waswas foundfound toto bebe greater under the full-braking condition than under the free-rolling condition. When the tyre load changedchanged fromfrom 2525 toto 4040 kN,kN, thethe verticalvertical stressstress ofof ththee free-rollingfree-rolling tyretyre increasedincreased fromfrom 11451145 toto 14251425 kPa,kPa, anan increaseincrease ofof 24.5%,24.5%, whilewhile thethe verticalvertical stressstress duringduring fullfull brakingbraking increasedincreased fromfrom 12431243 toto 16421642 kPa,kPa, anan increaseincrease ofof 32.1%.32.1%. ItIt isis evidentevident thatthat onceonce ththee overloadoverload isis excessiveexcessive duringduring fullfull braking,braking, thethe maximum value of vertical stress occurring at the tyre shoulder will increase the damage to the tyre.tyre. Sustainability 2020, 12, 10603 7 of 16

3.3. Effect of Tyre Load on the Change Characteristics of Contact Stresses The inflation pressure of 730 kPa and a tyre speed of 15 km/h were considered to be invariant, while the tyre load was varied to examine the effect of tyre load on tyre–pavement contact stresses. Figure7 shows the vertical contact stresses (CPRESS) under free rolling and full braking with tyre loads of 25 and 40 kN. With the gradual increase of the tyre load under the condition of steady rolling, the shape of the contact region was found to transition from approximately circular to rectangular, and the contact area also gradually increased. Regardless of whether the tyre was in the free-rolling or full-braking condition, the vertical stress at the tread of the tyre was found to change only slightly. However, the vertical stress was found to increase at the tyre shoulder. When the tyre load was small, the maximum value of vertical contact stress was near the centre of the tread, whereas when the load became large, the maximum value of vertical contact stress moved toward the tyre shoulder. Under different load conditions, the vertical stress was found to be greater under the full-braking condition than under the free-rolling condition. When the tyre load changed from 25 to 40 kN, the vertical stress of the free-rolling tyre increased from 1145 to 1425 kPa, an increase of 24.5%, while the vertical stress during full braking increased from 1243 to 1642 kPa, an increase of 32.1%. It is evident that once the overload is excessive during full braking, the maximum value of vertical stress occurring at the tyreSustainability shoulder 2020 will, 12, x FOR increase PEER REVIEW the damage to the tyre. 8 of 17

(a) (b)

Figure 7.FigureVertical 7. Vertical contact contact stresses stresses under under the free-rollingthe free-rolling condition condition with with tyre tyre loadsloads ofof (a) 25 kN kN and and (b) 40 (b) 40 kN; vertical contact stresses under the full-braking condition with tyre loads of (c) 25 kN and kN; vertical contact stresses under the full-braking condition with tyre loads of (c) 25 kN and (d) 40 kN. (d) 40 kN.

Figure8Figure shows 8 shows the longitudinal the longitudinal contact contact stresses stresses (CSHEAR1)(CSHEAR1) in in the the free-rolling free-rolling and full-braking and full-braking conditionsconditions with tyre with loads tyre loads of 25 of and25 and 40 40 kN. kN. The The shapes shapes of of the the contact contact region region under under free-rolling free-rolling and and full-brakingfull-braking conditions conditions were verywere diveryﬀerent. different. When When the the longitudinal longitudinal contact contact stress stresswas was calculatedcalculated under the full-brakingunder the condition, full-braking the condition, contact areathe contact shape andarea stressshape distributionand stress distribution were similar were to similar those whento the vertical stressthose when was calculated.the vertical stress However, was calculated. when the However, longitudinal when contact the longitudinal stress was contact calculated stress was under the calculated under the free-rolling condition, the shape of the contact region and stress distribution free-rolling condition, the shape of the contact region and stress distribution varied greatly from when varied greatly from when the vertical stress was calculated. According to Figure 8a,b, the the verticallongitudinal stress was contact calculated. stress had According positive and to Figurenegative8a,b, values the longitudinalduring free rolling, contact and stress the positive had positive and negativeand negative values values during were free relatively rolling, close; and this the ispositive mainly because and negative the torque values applied were to the relatively tyre was close; this is mainly0. It can because be seen thefrom torque Figure applied 8c,d that to the the longitudinal tyre was 0.contact It can stress be seen during from full Figure braking8c,d was that the longitudinalpositive. contact With stressthe increase during of full load, braking the longitudinal was positive. contact With stress the gradually increase ofincreased, load, the and longitudinal the contact stressposition gradually at which increased,the maximum and value the positionof longitud atinal which stress the occurred maximum also valuemoved of from longitudinal near the stress centre of the tread to the tyre shoulder, which was similar to the trend of vertical stress. As the tyre occurred also moved from near the centre of the tread to the tyre shoulder, which was similar to the load increased, the difference between longitudinal stress during full braking and that during free trend ofrolling vertical became stress. increasingly As the tyre larger. load This increased, illustrates thethat di overloadﬀerence under between in the longitudinalbraking condition stress not during full brakingonly anddoes that great during harm to free the rolling tyre but became can also increasingly accelerate the larger. emergence This illustratesof ruts and that cracks overload in the under pavement.

Sustainability 2020, 12, 10603 8 of 16 in the braking condition not only does great harm to the tyre but can also accelerate the emergence of Sustainability 2020, 12, x FOR PEER REVIEW 9 of 17 rutsSustainability and cracks 2020 in, 12 the, x FOR pavement. PEER REVIEW 9 of 17

(a) (b) (a) (b)

(c) (d) (c) (d) Figure 8. Longitudinal contact stresses under the free-rolling condition with tyre loads of (a) 25 kN FigureFigure 8. 8.Longitudinal Longitudinal contact contact stresses stresses under under the the free-rolling free-rolling condition condition with with tyre tyre loads loads of of (a ()a 25) 25 kN kN and and (b) 40 kN; longitudinal contact stresses under the full-braking condition with tyre loads of (c) 25 and (b) 40 kN; longitudinal contact stresses under the full-braking condition with tyre loads of (c) 25 (b)kN 40 and kN; ( longitudinald) 40 kN. contact stresses under the full-braking condition with tyre loads of (c) 25 kN andkN (d and) 40 ( kN.d) 40 kN. Figure 9 shows the lateral contact stresses (CSHEAR2) under free-rolling and full braking Figure 9 shows the lateral contact stresses (CSHEAR2) under free-rolling and full braking conditionsFigure9 showswith tyre the loads lateral of contact25 and stresses40 kN. (CSHEAR2)The shape of under the contact free-rolling region and for fullcalculating braking conditions with tyre loads of 25 and 40 kN. The shape of the contact region for calculating conditionstransverse with stress tyre was loads different of 25 andfrom 40 those kN. Thefor calc shapeulating of the vertical contact and region longitudinal for calculating contact transversestresses transverse stress was different from those for calculating vertical and longitudinal contact stresses stressunder was the di ﬀtwoerent working from those conditions. for calculating Transverse vertical contac andt stress longitudinal had positive contact and stressesnegative under values the that two under the two working conditions. Transverse contact stress had positive and negative values that workingwere close conditions. to each Transverse other under contact the two stress working had positive conditions. and negative This is valuesmainly thatbecause were during close to the each were close to each other under the two working conditions. This is mainly because during the otherlongitudinal under the movement two working process conditions. of the tyre, This the is mainlytyre did because not undergo during sideslip, the longitudinal and the transverse movement longitudinal movement process of the tyre, the tyre did not undergo sideslip, and the transverse processforce of the tyre tyre, was the close tyre didto the not equilibrium undergo sideslip, state. When and the tyre transverse load was force 25 kN, of the thetyre lateral was contact close to force of the tyre was close to the equilibrium state. When the tyre load was 25 kN, the lateral contact thestress equilibrium during state.free rolling When was the tyregreater load than was that 25 kN,during the lateralfull braking. contact When stress the during load freecontinued rolling to was stress during free rolling was greater than that during full braking. When the load continued to greaterincrease, than the that lateral during contact full braking. stresses When during the free load rolling continued and tofull increase, braking the were lateral closer, contact and stresses the increase, the lateral contact stresses during free rolling and full braking were closer, and the duringdifference free rolling between and them full became braking smaller. were closer, and the diﬀerence between them became smaller. difference between them became smaller.

(a) (b) (a) (b)

(c) (d) (c) (d) FigureFigure 9. 9.Lateral Lateral contact contact stresses stresses under under thethe free-rollingfree-rolling condition with with tyre tyre loads loads of of (a ()a 25) 25 kN kN and and (b) ( b) Figure 9. Lateral contact stresses under the free-rolling condition with tyre loads of (a) 25 kN and (b) 40 kN; lateral contact stresses under the full-braking condition with tyre loads of (c) 25 kN and (d) 4040 kN; kN; lateral lateral contact contact stresses stresses under under thethe full-brakingfull-braking condition condition with with tyre tyre loads loads of of(c) ( c25) 25 kN kN and and (d) ( d) 40 kN. 4040 kN. kN. Sustainability 2020, 12, 10603 9 of 16

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TakingTaking the tyrethe tyre load load of 32.5of 32.5 kN kN as as an an example, example, the spatial spatial distribution distribution of contact of contact stresses stresses under under the free-rollingthe free-rolling condition condition is shown is shown in Figure in Figure 10. 10. As As presented presented in Figurein Figure 10 b,10b, longitudinal longitudinal contact contact stress understress the free-rollingunder the free-rolling condition condition was symmetrically was symmetrically distributed distributed in a longitudinal in a longitudinal direction, direction, whereas, as presentedwhereas, in as Figure presented 10c, in the Figure lateral 10c, contact the latera stressl contact exhibited stress an almostexhibited antisymmetric an almost antisymmetric distribution in a longitudinaldistribution direction. in a longitudinal The longitudinal direction. contact The longitudinal stress was contact negative stress at was the frontnegative end at of the the front contact end area and positiveof the contact at the area back and end, positive and theat the distribution back end, and area the of distribution the negative area value of the of negative the contact value area of was smallerthe thancontact that area of was the positivesmaller than value. that of the positive value.

(a) (b)

(c)

FigureFigure 10. Spatial 10. Spatial distribution distribution characteristics characteristics of (aof) vertical,(a) vertical, (b) longitudinal,(b) longitudinal, and and (c) transverse(c) transverse contact stressescontact under stresses the free-rolling under the free-rolling condition condition with the with load the of load 32.5 of kN. 32.5 kN.

3.4. E3.4.ffect Effect of Inflation of Inflation Pressure Pressure on on the the Change Change Characteristics Characteristics of of Contact Contact Stresses Stresses The tyreThe loadtyre load of 25 of kN 25 and kN theand speed the speed of 15 of km 15/h km/h were were considered considered to be to invariant, be invariant, while while the inflationthe pressureinflation was pressure varied towas examine varied to the examine effect the of inflationeffect of inflation pressure pressure on tyre–pavement on tyre–pavement contact contact stresses. stresses. Table 1 presents maximum contact stresses under different inflation pressures under Table1 presents maximum contact stresses under di fferent inflation pressures under free-rolling and free-rolling and full-braking conditions. full-brakingAs the conditions. tyre inflation pressure was increased, the contact area decreased, and the vertical stress increased. Regardless of whether the tyres were in the free-rolling or full-braking condition, the verticalTable stress 1. wasComparison positive, indicating of maximum that contact the di stressesrection of under stress di ffwaserent downward. inflation pressures. It also indicates that the tyre tread was compressed and therefore bore compressive stress. To clearly observe the Inflation Maximum Contact Stress during Free Maximum Contact Stress during Full change trend, Figure 11 exhibitsRolling maximum (kPa) contact stresses under different Brakinginflation (kPa)pressures in the Pressureform of a colour histogram. (kPa) Vertical Longitudinal Lateral Vertical Longitudinal Lateral

530 931 194 240 1052 631 40 730 1145 142 303 1243 745 52 930 1386 86 381 1495 896 66 Sustainability 2020, 12, x FOR PEER REVIEW 11 of 17 Sustainability 2020, 12, 10603 10 of 16

Table 1. Comparison of maximum contact stresses under different inflation pressures.

As the tyre inflationMaximum pressure Contact was Stress increased, during the Free contact areaMaximum decreased, Contact and Stress the verticalduring Full stress Inflation increased. Regardless of whetherRolling the tyres (kPa) were in the free-rolling or full-brakingBraking condition, (kPa) the vertical Pressure (kPa) stress was positive,Vertical indicating thatLongitudinal the direction ofLateral stress wasVertical downward. Longitudinal It also indicates thatLateral the tyre tread530 was compressed931 and therefore 194 bore compressive240 stress.1052 To clearly631 observe the change40 trend,730 Figure 11 exhibits1145 maximum contact142 stresses under303 different1243 inflation pressures745 in the form 52 of a colour930 histogram. 1386 86 381 1495 896 66

(a) (b)

Figure 11. Maximum contactcontact stressesstresses underunder di differentﬀerent inﬂation inflation pressures pressures during during (a )( freea) free rolling rolling and and (b) full(b) full braking. braking.

When the inflationinflation pressure was 530 kPa, the maximummaximum vertical stress was located in the tyre shoulder. AsAs thethe inflation inflation pressure pressure increased increased from from 730 730 to 930 to kPa,930 kPa, the maximum the maximum vertical vertical stress understress theunder free-rolling the free-rolling condition condition increased increased from 1145 from to 13861145 kPa,to 1386 an increase kPa, an of increase 21.0%. Theof 21.0%. maximum The verticalmaximum stress vertical under stress the full-braking under the conditionfull-braking increased condition from increased 1243 to 1495 from kPa, 1243 an increaseto 1495 ofkPa, 20.3%, an andincrease the positionof 20.3%, of and the maximumthe position values of the moved maxi frommum the values tyre shouldermoved from to the the crown. tyre shoulder Under di fftoerent the inflationcrown. Under pressures, different the contactinflation area pressures, during thethe free-rollingcontact area condition during the was free-rolling similar for condition different tyrewas loads.similar When for different the inflation tyre loads. pressure When was the high, inflatio the magnituden pressure of was the high, positive the valuemagnitude of longitudinal of the positive stress wasvalue slightly of longitudinal less than thatstress of was the negativeslightly value.less than With that the of increase the negative of inflation value. pressure, With the longitudinal increase of stressinflation decreased, pressure, and longitudinal lateral stress stress increased. decreased, Under and the lateral full-braking stress increased. condition, Under the lateral the full-braking stress also increasedcondition, withthe lateral the increase stress also in tyre increased pressure, with but the it was increase not obvious. in tyre pressure, but it was not obvious. Taking the tyre pressure of 930 kPa as an example,example, the contact stress value of the node in the contact areaarea under under the full-brakingthe full-braking condition condition was extracted, was extracted, and the spatial and distributionthe spatial characteristics distribution ofcharacteristics the complex of contact the complex stresses contact under stresses a high tyreunder pressure a high tyre are exhibitedpressure are in Figureexhibited 12. in The Figure vertical 12. stressThe vertical under astress high tyreunder pressure a high was tyre greater pressure than was that greater under thethan basic that working under the condition, basic working and the spatialcondition, distribution and the wasspatial large distribution in the middle was and large small in on boththe middle sides. Longitudinaland small on contact both stresses sides. underLongitudinal the full-braking contact stresses condition under were the symmetrically full-braking condition distributed were in asymmetrically longitudinal direction,distributed while in a thelongitudinal lateral contact direction, stress presentedwhile the an lateral almost contact antisymmetric stress distributionpresented an in aalmost longitudinal antisymmetric direction. Indistribution addition, in the a frontlongitudinal end of thedirection. vertical In and addition, longitudinal the front stresses end of was the greater vertical than and the longitudinal back end, whichstresses was was determined greater than by thethe motion back end, state which of the tyre.was Overall,determined the lateralby the contact motion stress state was of lessthe tyre. than 100Overall, kPa andthe lateral therefore contact could stress almost was be less ignored. than 100 kPa and therefore could almost be ignored. SustainabilitySustainability2020 2020, 12, 12, 10603, x FOR PEER REVIEW 12 of11 17 of 16

(a) (b)

(c)

FigureFigure 12. 12.Spatial Spatial distribution distribution characteristics characteristics of (ofa) vertical,(a) vertical, (b) longitudinal,(b) longitudinal, and (andc) transverse (c) transverse contact stressescontact under stresses the under full braking the full conditionbraking cond withition inﬂation with inflation pressure pressure of 930 kPa.of 930 kPa.

3.5.3.5. Eff Effectect of of the the Tyre Tyre Speed Speed on on the the Change Change CharacteristicsCharacteristics of Contact Stresses Stresses TheThe tyre tyre load load of of 25 25 kN kN and and thethe inflationinflation pressure of of 730 730 kPa kPa were were considered considered to tobe beinvariant, invariant, whilewhile the the speed speed was was varied varied to to examine examine the the e effectffect of of speed speed onon tyre–pavementtyre–pavement contact stresses. stresses. Table Table 2 presents2 presents the the maximum maximum contact contact stresses stresses under under di differfferentent speedsspeeds during free free rolling rolling and and full full braking. braking. ToTo clearly clearly observe observe the the change change characteristics, characteristics, FigureFigure 13 showsshows the the maximum maximum contact contact stresses stresses under under diffdifferenterent inflation inflation pressures, pressures, in in the the formform ofof aa colourcolour histogram. As As the the speed speed increased, increased, the the vertical vertical andand longitudinal longitudinal stresses stresses gradually gradually decreased decreased under under the the free-rolling free-rolling condition, condition, but but the the reduction reduction was verywas small. very small. The lateralThe lateral stress stress remained remained essentially essentially unchanged. unchanged. UnderUnder the the full-braking full-braking condition, condition, verticalvertical stress stress increased increased slightly. slightly. TheThe longitudinallongitudinal stress stress remained remained basically basically unchanged, unchanged, and and the the transverse stress gradually increased, but the value of transverse stress was negligible compared transverse stress gradually increased, but the value of transverse stress was negligible compared with with the value of vertical stress. Compared with the tyre load and inflation pressure, the impact of the value of vertical stress. Compared with the tyre load and inflation pressure, the impact of tyre tyre speed on the maximum contact stresses is relatively weak. speed on the maximum contact stresses is relatively weak.

Table 2. Comparison of maximum contact stresses under different speeds. Table 2. Comparison of maximum contact stresses under diﬀerent speeds. Maximum Contact Stress During Free Rolling Maximum Contact Stress During Full Braking Speed Maximum Contact(kPa) Stress During Free Maximum Contact(kPa) Stress During Full Speed (km/h) Rolling (kPa) Braking (kPa) (km/h) Vertical Longitudinal Lateral Vertical Longitudinal Lateral 15 1145Vertical Longitudinal 142 Lateral 303 1243 Vertical Longitudinal 745 Lateral 52 45 1120 116 308 1254 752 82 15 1145 142 303 1243 745 52 90 1117 87 315 1256 753 175 45 1120 116 308 1254 752 82 90 1117 87 315 1256 753 175 Sustainability 2020, 12, x FOR PEER REVIEW 13 of 17

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(a) (b)

Figure 13. Maximum contact stresses under different speeds during (a) free rolling and (b) full braking. (a) (b)

4. EffectsFigure of 13.13. Tyre MaximumMaximum Working contact contact Conditions stresses stresses under onunder Rolling diff erentdifferent Resistance speeds speeds during during (a) free (a rolling) free rolling and (b) and full braking.(b) full braking. 4. EffDueects ofto Tyrethe viscoelasticity Working Conditions of tyre rubber on Rolling mate Resistancerial, the processes of compression and rebound are not the same in the rolling process, and therefore the sum of external forces on the tyre i.e., the 4. Effects of Tyre Working Conditions on Rolling Resistance RR, isDue not to 0 the[41]. viscoelasticity Figure 14 presents of tyre changes rubber material,in RR with the the processes tyre load, of inflation compression pressure, and rebound and speed. are notFigure theDue same14a to shows the in the viscoelasticity the rolling change process, in of RR andtyre with thereforerubber the tyre mate the load sumrial, when ofthe external processes the inflation forces of oncompression pressure the tyre i.e.,(P) andwas the RR, rebound730 is kPa, not 0areand [41 not ].the Figure the speed same 14 presents( inV) thewas rolling changes45 km/h. process, in RRWith and with the therefor the increase tyree load,the in sum inflationtyre of load,external pressure, tyre forces deformation and on speed. the tyre Figureincreased i.e., 14thea showsRR,significantly, is not the 0 change [41]. and Figure inthe RR overall with14 presents the RR tyre presented changes load when inan RR theupward with inflation the trend, tyre pressure load,which (inflationP )would was 730 pressure,cause kPa, the and andvehicle the speed. speed to (FigureconsumeV) was 14a 45 more kmshows/h. fuel. With the Figure change the increase 14b in RRshows in with tyre the load,the change tyre tyre load deformation in RRwhen with the increasedthe inflation pressure significantly, pressure when the(P and) wastyre the 730load overall kPa, (F) RRandwas presented 25the kN speed and an the(V upward) speedwas 45trend,(V )km/h. was which 45 With km/h. would the With causeincrease the the increase in vehicle tyre in toload, pressure, consume tyre tyredeformation more stiffness fuel. Figureincreased,increased 14b showssignificantly,and deformation the change and decreased inthe RR overall with under theRR pressure presentedthe same when load; an upward thethus, tyre the loadtrend, overall (F )which wasRR presen 25 would kNted and cause a thedownward speedthe vehicle (V trend,) was to 45consumewhich km /wouldh. more With result thefuel. increase inFigure the vehicle in14b pressure, shows consuming the tyre change stilessffness fuel. in increased,RR Figure with 14cthe and showspressure deformation the whenchange decreasedthe in tyre RR withload under the(F) thewastyre same 25load kN load;and and speed thus, the speed thewhen overall ( Vthe) was pressure RR 45 presented km/h. (P) wasWith a downward 730 the kPa, increase and trend, itin was whichpressure, found would tyre that resultstiffness the increased in theincreased, vehicle tyre consumingandload deformation led toless increased fuel. decreased Figure RR. 14 Moreover,underc shows the the sameas changethe load; speed in th RRus, increased, with the overall the tyre the RR load loss presen and factor speedted ofa whendownward tyre therolling pressure trend, and (whichdeformationP) was would 730 kPa, decreased result and in it the was[42]. vehicle found However, consuming that thethe increased increase less fuel.d tyre value Figure load of led14c RR toshows was increased greater the change RR. than Moreover, inthe RR decreased with as the speedtyrevalue, load increased,so theand overall speed the loss RRwhen still factor the increased. ofpressure tyre rolling (P) andwas deformation730 kPa, and decreased it was found [42]. However,that the increased the increased tyre valueload led of RRto wasincreased greater RR. than Moreover, the decreased as the value, speed so theincreased, overall RRthe stillloss increased. factor of tyre rolling and deformation decreased [42]. However, the increased value of RR was greater than the decreased value, so the overall RR still increased.

(a) (b)

Figure 14. Cont. (a) (b) SustainabilitySustainability2020 2020,,12 12,, 10603x FOR PEER REVIEW 1413 of 16 17

(c)

FigureFigure 14.14. RollingRolling resistance (RR) changes with (a)) tyretyre load,load, ((bb)) inﬂationinflation pressure,pressure, andand ((cc)) tyretyre loadload andand speed.speed.

TheThe exponentialexponential EquationEquation (8) (8) is is used used to fitto thefit th functionale functional relationship relationship between between RR and RR tyreand load,tyre inflationload, inflation pressure, pressure, and speed andparameters, speed parameters, and variable and combinationsvariable combinations were considered, were considered, as shown inas Figureshown 14 in. Figure The exponential 14. The exponential equation is equation as follows: is as follows: 2 mn RRabVcVFP=+⋅+⋅⋅() ⋅ (8) RR = (a + b V + c V2) Fm Pn (8) where a, b, and c are velocity correlation coefficients,· · m· is· the tyre load exponent, and n is the whereinflationa, b ,pressure and c are exponent. velocity correlation According coe ffitocients, these msetsis the of tyrevariable load exponent,combinations, and n isthe the following inflation pressurerelationship exponent. is obtained According by fitting to the these functional sets of variable relationship: combinations, the following relationship is 2 1.13258− 0.28753 obtained by fittingRR the functional=−⋅+⋅⋅⋅(0.13153 relationship: 0.00211 V 0.00001 V ) F P (9) The international standard unit was adopted for the unit of each parameter in the process of 2 1.13258 0.28753 RR = (0.13153 0.00211 V + 0.00001 V ) F P− (9) fitting. The parameters in Equation− (9) are different· for ·different· types· of tyres, but Equation (9) describes a method that can calculate the RR related to tyre working conditions. The tyre studied in this articleThe international is one of the standard most commonly unit was adopted used truc fork-bus the unit tyres of eachin China. parameter Therefore, in the processEquation of (9) fitting. can Thebe used parameters to forecast in Equation the RR (9) of are the di fftyreerent studied for diff erentin this types article of tyres,in any but combination Equation (9) of describes working a methodconditions, that and can a calculate similar method the RR relatedcan be used to tyre to workingforecast the conditions. RRs of other The types tyre studied of tyres. in this article is one of the most commonly used truck-bus tyres in China. Therefore, Equation (9) can be used to forecast5. Conclusions the RR of the tyre studied in this article in any combination of working conditions, and a similar method can be used to forecast the RRs of other types of tyres. The tyre-pavement contact model developed in this work shows the potential to predict 5.contact Conclusions stresses and rolling resistance under different rolling conditions. Based on the preceding analysis, the following conclusions can be made and provide valuable suggestions and viewpoints The tyre-pavement contact model developed in this work shows the potential to predict contact for tyre–pavement contact mechanics and fuel-economic tyre production/design. stresses and rolling resistance under different rolling conditions. Based on the preceding analysis, the(1) followingThe maximum conclusions value of can the be transverse made and contact provide stre valuabless is greater suggestions than the andlongitudinal viewpoints contact for tyre–pavementstress under contact the free-rolling mechanics condition. and fuel-economic However, tyre the production situation under/design. the full-braking condition is the opposite. (1)(2) LongitudinalThe maximum contact value of thestresses transverse under contact the stressfree-rolling is greater and than full-braking the longitudinal conditions contact stresswere symmetricallyunder the free-rolling distributed condition. in a longitudinal However, direction, the situation while under the lateral the full-braking contact stress condition presented is anthe opposite.almost antisymmetric distribution in a longitudinal direction. Under the full-braking (2) condition,Longitudinal longitudinal contact stresses stress under is the the main free-rolling component and of full-braking the horizontal conditions contact were stresses. symmetrically (3) Thedistributed tyre load in and a longitudinal inflation pressure direction, have while significant the lateral impacts contact on contact stress stresses. presented Overload an almost and lowantisymmetric tyre pressure distribution are important in a longitudinal contributorsdirection. to the wear Under of the the tyre full-braking shoulder. condition, Properly increasinglongitudinal the stress inflation is the pressure main component can effectively of the relieve horizontal damage contact to stresses.the tyre shoulder caused by (3) overloading.The tyre load Additionally, and inflation pressurecompared have with significant the tyre load impacts and on inflation contact pressure, stresses. Overloadthe impact and of tyrelow tyrespeed pressure on contact are important stress is relatively contributors weak. to the wear of the tyre shoulder. Properly increasing (4) Thethe inflationproposed pressure exponential can e ffequationectively relievedescribes damage a method to the that tyre can shoulder forecast caused the RR by related overloading. to the working conditions of truck-bus tyres, and a similar method can be used to predict the RRs of other types of tyres. Sustainability 2020, 12, 10603 14 of 16

Additionally, compared with the tyre load and inﬂation pressure, the impact of tyre speed on contact stress is relatively weak. (4) The proposed exponential equation describes a method that can forecast the RR related to the working conditions of truck-bus tyres, and a similar method can be used to predict the RRs of other types of tyres.

It is evident from the full analysis presented in this study that the change rules of contact stresses and RR are significantly affected by tyre working conditions under different rolling conditions. Conventional methods focus on the RR of patterned tyres of passenger cars or on the adoption of special materials to reduce the RR of truck tyres. Our new approach considers the different working conditions of the tyre by investigating changes in contact stresses (forces) to pinpoint the functional relationship between the RR and working conditions of the truck-bus tyre. The proposed exponential equation method can be usefully applied to predict the RR, and better estimations of fuel consumption and greenhouse gas emissions can then be made.

Author Contributions: Conceptualization, M.G. and X.L.; methodology, M.G.; software, M.G.; validation, M.G. and X.L.; formal analysis, M.G. and X.L.; investigation, M.G. and X.L.; resources, M.G. and X.L.; data curation, M.G. and X.L.; writing—original draft preparation, M.G. and X.L.; writing—review and editing, M.R. and Y.Y.; visualization, M.R. and Y.Y.; supervision, X.Z.; project administration, M.G. and X.L.; All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Major Scientific Research Instrument Development Project of China (No. 51827812), National Natural Science Foundation of China (Nos. 51778509 and 51578430). Conflicts of Interest: The authors declare that there is no conflict of interest.

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